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Application of Dimensional analysis (V)- As a research tool to derive new relations - (Concept)
The method of dimensional analysis can be used to derive new relations.
For example, we can derive a relation for the Time period of a simple pendulum.
$
T=K m^a l^b g^c
$
where
$
T=\text { time period }
$
$l=$ length
$g=$ acceleration due to gravity
So
Equating exponents of similar quantities
$
a=0 b=1 / 2 c=-1 / 2
$
We get
$
\therefore T=2 \pi \sqrt{\frac{l}{g}}
$
| Exam | Chapter |
| JEE MAIN | Physics and Measurement |
Critical velocity depends upon the coefficient of viscosity, density and radius and is given by the relation:
, where
Find value of x,y,z
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The dimension of stopping potential $V_0$ in photoelectric effect in units of Planck's constant ' h ', speed of light ' c ', gravitational constant ' G ' and ampere A is :
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A metal sample carrying a current along the X-axis with density Jx is subjected to a magnetic field Bz (along the Z-axis). The electric field Ey developed along the Y-axis is directly proportional to Jx as well as Bz. The constant of proportionality has an SI unit.
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If the time period t of the oscillation of a drop of liquid of density d, radius r, vibrating under surface tension s is given by the formula . It is observed that the time period is directly proportional to
, The value of b should therefore be :
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If momentum P, area A, and time T are taken to be fundamental quantities, then the dimensional formula for energy is:-
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Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Time period of oscillation of a liquid drop depends on surface tension (S), if
the density of the liquid is and the radius of the drop is
, then
is
dimensionally correct, where is dimensionless.
Reason (R): Using dimensional analysis we get R.H.S. having different dimensions than that of time period.
In the light of above statements, choose the correct answer from the options given below.
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If time , and angular momentum
are taken as the fundamental units. Then the dimension of mass
in terms of
and
is :
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If force (F), length (L), and time (T) are taken as the fundamental quantities, then what will be the dimension of density?
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In a system of units of force (F), acceleration (A) and time (T) are taken as fundamental units, then the dimensional formula of energy is :
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To find the value of ‘g’ using a simple pendulum. T = 2.00 sec; l = 50.0 cm was measured. The maximum permissible error in ‘g’ is :
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If electronic charge e, electron mass m, speed of light in vacuum c and Planck’s constant h are taken as fundamental quantities, the permeability of vacuum can be expressed in units of:
| A. |
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| B. |
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Expression for time in terms of G (universal gravitational constant), h (Planck constant) and c (speed of light) is proportional to:
| A. |
|
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| D. |
|
The ratio of the dimensions of Plank's constant and that of the moment of inertia is the dimension of :
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A physical quantity Q depends on the variables as
where
is a constant of proportionality. If the dimensions of
are L, M and T, respectively, then the dimensions of
in terms of L, M and T are :
| A. |
|
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| D. |
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The torque produced by a force F acting at a distance r from the axis of rotation is given by
, where
is the angle between the force vector and the radius vector. Using dimensional analysis, find the relationship between torque, force, and distance.
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| C. |
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| D. |
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If the velocity of light c, universal gravitational constant G and Planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system are:
| A. |
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| D. |
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A force is represented by $F=a x^2+b t^{\frac{1}{2}}$ where $x=$ distance and $t=$ time. The dimensions of $b^2 / a$ are :
| A. |
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|
Consider two physical quantities A and B related to each other as $E=\frac{B-x^2}{A t}$ where E, x and t have dimensions of energy, length and time respectively. The dimension of A B is:
| A. |
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Time period of simple pendulum T depends on the mass (m), effective length (l) and acceleration due to gravity (g)
As T = K mx 2y gZ
Where K is dimension less constant.
The value of x y and Z are?
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|
The dimensions of Stefan-Boltzmann constant $\sigma$ can be written in terms of Plank's constant $h$, Boltzmann constant $k_b$ and the speed of light $c$ as $\sigma=h^\alpha k_b^\beta c^\gamma$. Here
| A. |
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The dimensions of the area $A$ of a black hole can be written in terms of the universal gravitational constant $G$, its mass $M$, and the speed of light c as $A=G^\alpha M^\beta c^\gamma$. Here
| A. |
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If a quantity Q is given by $Q=\frac{M F \eta G}{Y T^2}$ where $M=$ mass, $F=$ force, G=gravitational constant, $\eta=$ Coefficient of viscosity, $\mathrm{Y}=\mathrm{Young}$ 's modulus. Then what is the dimension of Q in terms of $\mathrm{P}, \varepsilon_0, \mu_0$ where $\varepsilon_0=$ permittivity of free space and $\mu_0=$ permeability of free space and $\mathrm{P}=$ Power?
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The real gas equation is given as:
$\left(P+\frac{a}{V^2}\right) \cdot(V-b)=$ constant.
Then dimension of $\frac{a}{b^2}$ is the same as that of:
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Young-Laplace law states that the excess pressure inside a soap bubble of radius R is given by $\Delta P=4 \sigma / R {\text { where }} \sigma$ is the coefficient of surface tension of the soap. The Eotvos number $E_0$ is a dimensionless number that is used to describe the shape of bubbles rising through a surrounding fluid. It is a combination of g, the acceleration due to gravity p, the density of the surrounding fluid, $\sigma$ and a characteristic length scale $L$ which could be the radius of the bubble. A possible expression for $E_0$ is
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The earth's magnetic fields were flipped by $180^{\circ}$ a million years ago. This flip was relatively rapid and took $10^5$ years. Then the average change in orientation per year during the flip was closest to:
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Stoke's law states that the viscous drag force F experienced by a sphere of radius a, moving with a speed v through a fluid with a coefficient of viscosity $\eta$ , is given by
$
F=6 \pi \eta a v
$
If this fluid is flowing through a cylindrical pipe of radius r, length $l$ and a pressure difference of P across its two ends, then the volume of water $V$ which flows through the pipe in time $t$ can be written as
$
\frac{v}{t}=k\left(\frac{p}{l}\right)^a \eta^b r^c
$where k is a dimensionless constant. Correct values of $\mathrm{a}, \mathrm{b}$ and c are:
| A. |
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| C. |
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| D. |
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In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of $\mathrm{M}^{\mathrm{P}} \mathrm{L}^{\mathrm{Q}} \mathrm{T}^{\mathrm{R}} \mathrm{A}^{\mathrm{S}}$, where value of ' Q ' and ' R ' are
| A. |
|
| B. |
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| C. |
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| D. |
|
In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic dipole moment has dimension of $\left[M^P L^Q T^R A^S\right]$. The value of $P$ and $Q$ are :
| A. |
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| B. |
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| C. |
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| D. |
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If the dimension of $[\mu_0]=K$ and the dimension of $\left[\varepsilon_0\right]=P$ then what is the dimension of the speed of light in terms of P and K :
| A. |
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| D. |
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If P = Power and W = work then which of the following Quantity has a dimension equal to [T]
| A. |
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| D. |
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Velocity and acceleration
in two systems of units 1 and 2 are related as
and
respectively. Here m and n are constants. The relations for distance and time in the two systems respectively are:
| A. |
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| B. |
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| C. |
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| D. |
|
A point particle is acted upon by a restoring force $-k x^3$. The time period of oscillation is $T$ when the amplitude is A. The time period for an amplitude 2 a will be
(A) T
(B) $\mathrm{T} / 2$
(C) 2 T
(D) 4 T
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| D. |
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